Fundamentals of Signal Processing

Discrete-Time Signals and Systems

Mathematically, analog signals are functions having as their
independent variables continuous quantities, such as space and
time. Discrete-time signals are functions defined on the
integers; they are sequences. As with analog signals, we seek
ways of decomposing discrete-time signals into simpler
components. Because this approach leading to a better
understanding of signal structure, we can exploit that structure
to represent information (create ways of representing
information with signals) and to extract information (retrieve
the information thus represented). For symbolic-valued signals,
the approach is different: We develop a common representation of
all symbolic-valued signals so that we can embody the
information they contain in a unified way. From an information
representation perspective, the most important issue becomes,
for both real-valued and symbolic-valued signals, efficiency:
what is the most parsimonious and compact way to represent
information so that it can be extracted later.

Real- and Complex-valued Signals

A discrete-time signal is represented symbolically as
sn, where
n…-101….

Cosine

The discrete-time cosine signal is plotted as a stem
plot. Can you find the formula for this signal?
We usually draw discrete-time signals as stem plots to
emphasize the fact they are functions defined only on the
integers. We can delay a discrete-time signal by an integer
just as with analog ones. A signal delayed by m samples has the
expression
snm.

Complex Exponentials

The most important signal is, of course, the complex
exponential sequence.

sn2fn

Note that the frequency variable f is dimensionless and that adding an integer to the frequency of the discrete-time
complex exponential has no effect on the signal's value.

2fmn2fn2mn2fn

This derivation follows because the complex exponential
evaluated at an integer multiple of
2 equals one. Thus, we need only consider frequency to have a value in some unit-length interval.

Sinusoids

Discrete-time sinusoids have the obvious form
snA2fnφ. As opposed to analog complex exponentials and
sinusoids that can have their frequencies be any real value,
frequencies of their discrete-time counterparts yield unique
waveforms only when
f lies in the interval
1212. This choice of frequency interval is arbitrary; we can also choose the frequency to lie in the interval
01.
How to choose a unit-length interval for a sinusoid's frequency will become evident later.

Unit Sample

The second-most
important discrete-time signal is the unit sample,
which is defined to be

δn1n00

Unit sample

The unit sample.
Examination of a discrete-time signal's plot, like that of the
cosine signal shown in [link],
reveals that all signals consist of a sequence of delayed and
scaled unit samples. Because the value of a sequence at each
integer
m is denoted by
sm and the unit sample delayed to occur at
m
is written
δnm, we can decompose any signal as
a sum of unit samples delayed to the appropriate location and
scaled by the signal value.

snmsmδnm

This kind of decomposition is unique to discrete-time signals,
and will prove useful subsequently.

Unit Step

The unit sample in discrete-time is well-defined at the origin, as opposed to the situation with analog signals.

un1n00n0

Symbolic Signals

An interesting aspect of discrete-time signals is that their
values do not need to be real numbers. We do have real-valued
discrete-time signals like the sinusoid, but we also have
signals that denote the sequence of characters typed on the
keyboard. Such characters certainly aren't real numbers, and
as a collection of possible signal values, they have little
mathematical structure other than that they are members of a
set. More formally, each element of the
symbolic-valued signal
sn
takes on one of the values
a1…aK
which comprise the alphabetA. This technical terminology does
not mean we restrict symbols to being members of the English
or Greek alphabet. They could represent keyboard characters,
bytes (8-bit quantities), integers that convey daily
temperature. Whether controlled by software or not,
discrete-time systems are ultimately constructed from digital
circuits, which consist entirely of
analog circuit elements. Furthermore, the transmission and
reception of discrete-time signals, like e-mail, is
accomplished with analog signals and systems. Understanding
how discrete-time and analog signals and systems intertwine is
perhaps the main goal of this course.

Discrete-Time Systems

Discrete-time systems can act on discrete-time signals in ways
similar to those found in analog signals and systems. Because
of the role of software in discrete-time systems, many more
different systems can be envisioned and "constructed" with
programs than can be with analog signals. In fact, a special
class of analog signals can be converted into discrete-time
signals, processed with software, and converted back into an
analog signal, all without the incursion of error. For such
signals, systems can be easily produced in software, with
equivalent analog realizations difficult, if not impossible,
to design.